Symmetric-group representations and Young diagrams

A partition of n is a decreasing list of positive row lengths summing to n. Its Young diagram has those left-justified rows. A tableau fills the boxes with the numbers one through n without repetition. A standard tableau has entries increasing across every row and down every column. Different fillings of a fixed shape do not label different complex irreducible types; they help construct or index vectors associated with that same type. The distinction between a shape and a filling is essential when counting dimensions.

Fix one filling and consider permutations preserving each row as a set, and permutations preserving each column as a set. Sum the row permutations without signs to get a row symmetrizer, and sum the column permutations with their signs to get a column antisymmetrizer. Their product is a Young symmetrizer, with its order declared because the factors generally do not commute. We use unnormalized sums and put the row factor on the left. The left ideal generated by that element is preserved by left multiplication by every permutation.

The complex Specht-module classification says each left ideal just constructed is irreducible, every complex irreducible of the symmetric group arises in this way, and two shapes give equivalent modules exactly when their partitions agree. Changing the filling changes the presentation but preserves the type. We invoke this theorem rather than presenting its full straightening and irreducibility proof. The standard-basis theorem further identifies the dimension with the number of standard tableaux of the shape, providing a combinatorial meaning for a matrix dimension.

A single row gives the trivial representation because all labels are symmetrized, and a single column gives the sign representation because all labels are antisymmetrized. Other shapes mix these constraints in subtler ways. In characteristic dividing the group order, these same constructions require different irreducibility statements: Specht modules can be reducible, and partitions do not give the same simple one-to-one classification of irreducibles. The complex theorem cannot simply be transferred by reducing its matrices modulo a prime. Maschke's characteristic restriction explains part of that difficulty.

The hook length counts only the arm to the right and the leg below, not an entire southeast rectangle. Computing each box separately avoids multiplying incorrect row lengths or counting the corner twice. For a single row the hook product is the factorial, giving dimension one; a single column gives the same result. Transposing any shape preserves its hook multiset and therefore its dimension, though the corresponding representation need not be equivalent. In fact transposition corresponds to tensoring with the sign representation over the complex numbers.

The group order supplies another independent check. List all partitions of a small n, compute their hook dimensions, and sum the squares. Completeness of the complex classification requires that sum to equal the factorial. For larger n the hook formula is far more efficient than constructing all permutations, but it still does not provide every character value at other conjugacy classes. Diagram combinatorics controls several different quantities, and a dimension formula should not be mistaken for a full character formula or an explicit basis of matrices.